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What is the Fibonacci sequence? Learn about the Fibonacci sequence definition, the golden ratio in nature, the Fibonacci spiral, and Fibonacci sequence examples.Updated: 11/21/2023
Frequently Asked Questions
Why is the Fibonacci sequence so important?
The Fibonacci sequence is important for many reasons. In nature, the numbers and ratios in the sequence can be found in the patterns of petals of flowers, the whorls of a pine cone, and the leaves on stems. As the sequence continues, the ratios of the terms approach a number known as the golden ratio. This ratio is prominent in architecture and works of art as well. As the ratios approach the golden ratio, they form a spiral know as the golden spiral. This spiral is found in many natural phenomena such as the nautilus, the spiral galaxies, and the formation of many flowers.
Why is the golden ratio important in nature?
The golden ratio is important in nature, because it naturally occurs in many ways in nature. Some examples are the way seashells grow, the scales of a pine cone, and the ratio of the number of leaves on a stem. Because it occurs so often, the golden ratio is sometimes called the "Divine Proportion."
What is the Fibonacci sequence and how does it works?
The Fibonacci sequence is the sequence of numbers 1, 1, 2, 3, 5, 8, 13, 21, 34, and so on. Any term of the sequence can be determine by adding the two terms that came before it. That is, the sequence is defined by starting with the terms 1 and 1, then adding the two previous terms to get the next term. For example 1 + 1 = 2, then 1 + 2 = 3, 2 + 3 = 5, 3 + 5 = 8, and so on. This pattern continues to generate the Fibonacci sequence of numbers 1, 1, 2, 3, 5, 8, 13, . . .
Table of Contents
- What is the Fibonacci Sequence?
- Fibonacci Sequence Examples
- The Fibonacci Spiral
- What is the Golden Ratio?
- Uses of the Fibonacci Sequence
- Lesson Summary
The Fibonacci sequence is the sequence of numbers given by 1, 1, 2, 3, 5, 8, 13, 21, 34, and so on. Each term of the sequence is found by adding the previous two terms together. The Fibonacci sequence must start with the first two terms being 1 and 1. The mathematical Fibonacci sequence definition uses the following rules.
- {eq}a_{1} = 1 {/eq}
- {eq}a_{2} = 1 {/eq}
- {eq}a_{n}=a_{n-1}+a_{n-2} {/eq} for n {eq}\geq {/eq} 3
Where {eq}a_i {/eq} is the {eq}i^{\text{th}} {/eq} term of the sequence.
It is useful to see how the terms of the sequence relate using a table.
| an-2 | an-1 | an-2 + an-1 = an |
|---|---|---|
| 1 | 1 | 2 |
| 1 | 2 | 3 |
| 2 | 3 | 5 |
| 3 | 5 | 8 |
| 5 | 8 | 13 |
| 8 | 13 | 21 |
| 13 | 21 | 34 |
We see that each term of the sequence is found by adding the two preceding terms together. In mathematics, we call this type of a sequence, in which the terms of the sequence are derived from previous terms, a recursive sequence.
In mathematics there are different kinds of sequences that are defined based on the relationship of the terms. In an arithmetic sequence, the same number is added to each term to get the next term. In a geometric sequence, a constant is multiplied by each term to get the next term. However, the Fibonacci sequence can only be generated by using the previous two terms to get the next term, therefore it is a recursive sequence.
The Origin of the Fibonacci Sequence
Leonardo of Pisa was born in Italy around 1170. He was the son of an Italian businessman and also called Fibonacci which means "son of Bonacci." Italian merchants needed to make a lot of mathematical calculations during trades at that time and were using the Roman Numeral system to do so. When Fibonacci learned the Hindu-Arabic system of numbers, he realized that the arithmetic used in this system was easier than using the Roman Numeral system, and he published these findings in his book "Liber Abachi."
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Fibonacci's book also included a mathematical problem focused on the reproduction of rabbits. Fibonacci's scenario goes as follows:
- Start with one pair of male and female rabbits .
- After the rabbits are one month old, they can mate and produce one pair of male and female rabbits.
- Newly born rabbits wait one month and then that pair will produce a new pair of rabbits.
- The rabbits of mating age continue to produce one pair of new rabbits every month.
- The number of total pairs of rabbits each month results in the Fibonacci sequence.
This scenario is pictured in the diagram.
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In addition to the pattern that generates the sequence, the terms of the Fibonacci sequence contain patterns within themselves. The individual numbers in the Fibonacci sequence list are referred to as Fibonacci numbers or Fibonacci sequence numbers. A ratio comparing two consecutive Fibonacci numbers in the sequence is called a Fibonacci ratio, for example 3:5 or 21:13 are Fibonacci ratios, because they compare a Fibonacci number to the Fibonacci number that comes before or after it in the sequence. It was discovered that the Fibonacci numbers and ratios occur in many places in the real world, such as:
- shells
- flowers and fruits
- architecture
For example, many flowers have a Fibonacci number as their number of leaves or petals, such as 3, 5, 8, or 13. The lily has 5 petals, some daisies have 13 petals, and a chicory has 21 petals.
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The Fibonacci Spiral is formed by starting with a square of side length of 1, then creating squares with the side lengths of the rest of the Fibonacci numbers and placing them geometrically together in a systematic fashion. Arcs are then drawn to connect certain points of the squares, and this result in the spiral that we call the Fibonacci Spiral. The pattern is best seen with a diagram.
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The Fibonacci Spiral in Nature
The Fibonacci Spiral is seen in nature in many ways such as the shape of a nautilus (seashell), the arrangement of the spirals of a sunflower, and the arrangement of the scales of a pinecone.
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The golden ratio is defined to be the number {eq}\dfrac{(1+\sqrt{5})}{2}=1.6180339... {/eq}, and is represented by the symbol {eq}\phi {/eq}. The approximation of the golden ratio is 1.618. The golden ratio was studied as early as 300 BC by Euclid and by mathematicians and scientists throughout history. Kepler observed the ratio in the solar system, and other scientists observed the ratio in geology and physics. The relationship between {eq}\phi {/eq} and the Fibonacci sequence was not discovered until the late 1600's. This relationship is that the Fibonacci ratio of a Fibonacci number to the previous Fibonacci number approaches the golden ratio as the sequence continues. This relationship can be seen in the table.
| Fibonacci term | Previous Term | Ratio |
|---|---|---|
| 2 | 1 | 2 |
| 3 | 2 | 1.5 |
| 5 | 3 | 1.6666 |
| 8 | 5 | 1.6 |
| 13 | 8 | 1.625 |
| 21 | 13 | 1.6153 |
| 34 | 21 | 1.6190 |
| 55 | 34 | 1.6176 |
| 89 | 55 | 1.618 |
This pattern can be seen geometrically in the image.
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The Golden Ratio in Nature
Just as Fibonacci numbers and the Fibonacci spiral are evident in nature, so is the golden ratio since all three of the mathematical concepts are intertwined. Because of its frequent occurrence in nature, the golden ratio is often called the "Divine Proportion." The spiral of many objects in nature have ratios that approach the golden ratio. Some examples are a snail's shell, the spiral aloe, a spiral galaxy, spider webs, and the Folha.
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In addition to nature, the Fibonacci sequence and the golden ratio are apparent in other areas such as:
- architecture
- art
- computer science
Mathematicians as early as Pythagoras suggested that dimensions using the Fibonacci ratios were aesthetically pleasing, and therefore many works of art and architecture are created with these ratios. Some examples are the proportions that contain Fibonacci ratios found in the rectangular structures of the Parthenon and the dome of the Santa Maria del Fiore Cathedral in Florence.
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Proportions in many paintings and sculptures are said to be based on Fibonacci ratios. Some notable examples are the Mona Lisa, Da Vinci's Vitruvian Man, and Polycl*tus' statue Doryphorus. Da VInci was said to use the Fibonacci ratios as dimensions to construct rectangles to organize his artwork. The Mona Lisa contains rectangles that frame the image with dimensions that are Fibonacci numbers and form the golden ratio.
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Computer science languages are based on binary code, which is a sequence of 0's and 1's that execute commands. There is a way to use the Fibonacci representation of a number to create binary code. The algorithm using the Fibonacci representation can be used with a variety of programming languages.
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The Fibonacci Sequence was discovered by Leonardo of Pisa in the 12th century. Also known as Fibonacci, he explained the sequence through a story about rabbits mating.
- The terms of the sequence are 1, 1, 2, 3, 5, 8, 13, 21, 34 and so on.
- Each term is found by adding the previous two terms.
- Each term is called a Fibonacci number.
- The ratios comparing consecutive terms of the Fibonacci sequence are called Fibonacci ratios.
Fibonacci numbers and Fibonacci ratios are found frequently in nature. Some examples are the number of petals on flowers, the ratio of the whorls on a pine cone, and leaves on the stems of a flower. When squares with side lengths equal to the Fibonacci numbers are placed together geometrically they form the Fibonacci spiral. This spiral is also found in nature. Some examples of the spiral are found in the arrangement of the seeds on the head of a sunflower and in the nautilus, or seashell.
- As the ratios of each term of the Fibonacci sequence to the previous term progress, the ratio approaches the irrational number 1.618....
- This irrational number is call the golden ratio and is exactly equal to {eq}\dfrac{ 1 + \sqrt{5}}{2}. {/eq}
- The golden ratio is also seen in nature in spiral galaxies, spider webs, and plants.
- The Fibonacci sequence, the Fibonacci ratios, and the golden ratio are all intertwined.
- The Fibonacci sequence and the golden ratio are also used in architecture, art, and computer science.
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Video Transcript
Definition
The Fibonacci sequence begins with the numbers 0 and 1. The third number in the sequence is the first two numbers added together (0 + 1 = 1). The fourth number in the sequence is the second and third numbers added together (1 + 1 = 2). Each successive number is the addition (the sum) of the previous two numbers in the sequence. The sequence ends up looking like this:
0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, and so on and so forth.
Looking at it, you can see that each number in the sequence is the addition or sum of the two previous numbers. For example, 34 is the addition of 21 and 13. 144 is the addition of 89 and 55. Try it out yourself and check other numbers in the sequence to see if they follow the rule.
The Golden Ratio
The golden ratio, represented by the Greek letter phi, is approximately 1.618. The golden ratio, like pi, is an irrational number that keeps going. The actual value goes like this: 1.618033988764989. . .
You might be wondering how the Fibonacci sequence relates to this number. Let us see.
Let's start by dividing pairs of numbers in the Fibonacci sequence. We will skip zero and start with the pair of ones. 1 / 1 = 1. The next pair is the one and the two. 2 / 1 = 2. In each pair, we divide the larger by the smaller number. Let's keep going and see where it takes us:
| Fibonacci pair | Result |
|---|---|
| 2 and 3 | 3 / 2 = 1.5 |
| 3 and 5 | 5 / 3 = 1.6666. . . |
| 5 and 8 | 8 / 5 = 1.6 |
| 8 and 13 | 13 / 8 = 1.625 |
| 13 and 21 | 21 / 13 = 1.6154. . . |
| 21 and 34 | 34 / 21 = 1.619. . . |
| 34 and 55 | 55 / 34 = 1.618. . . |
| 55 and 89 | 89 / 55 = 1.618. . . |
| 89 and 144 | 144 / 89 = 1.618. . . |
As the numbers get larger, an interesting thing starts to happen. The result of dividing the pairs of numbers gives you the approximate value of the golden ratio, 1.618. . .
In mathematical terms, the Fibonacci sequence converges on the golden ratio. What that means is that, as the Fibonacci sequence grows, when you divide a pair of numbers from the sequence, the result will get closer and closer to the actual value of the golden ratio. Looking at the table, you can see that starting with the pair 34 and 55, the result is accurate to three decimal places. As the pairs get larger, the result will be more accurate to the decimal.
Nature
The Fibonacci sequence approximates the golden ratio, which can be found in the natural world. You can see it in your own body, in the way seashells grow, and the number of petals in flowers. Let's take a look at these real-life examples.
Take a look at your own fingers. Notice that each finger has three parts to it. If you measure each section and divide pairs of sections, you will get an approximate value of the golden ratio. If the smallest section of your finger measures one unit, then the section below will measure roughly two units, and the third section will measure about three units. Notice how each measurement corresponds to a Fibonacci number.
Seashells grow in a Fibonacci sequence. If you tile squares with sizes that follow the Fibonacci sequence (1, 1, 2, 3, 5, and so on) and draw a spiral that connects to each outer edge, you will see a seashell forming.
You will also see the Fibonacci numbers in the way flowers grow their petals. An orchid, for example, has several layers of petals and each layer corresponds to a Fibonacci number. Look at the orchid picture, and you will see a petal layer of two, then a petal layer of three, followed by an outer petal layer of 5.
Lesson Summary
To summarize, the Fibonacci sequence begins with 0 and 1, and each successive number is the sum of the two previous numbers. As the Fibonacci sequence grows, if you divide pairs of numbers in the sequence (the larger by the smaller), you will get an approximate value of the golden ratio, which is roughly 1.618.
Learning Outcomes
When you are finished, you should be able to:
- Describe and identify the Fibonacci sequence
- Explain the relationship between the Fibonacci sequence and the golden ratio
- Recall some examples of the Fibonacci sequence in nature
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